Stop Calculating the "Whole" — Kill O(N^2) for a 31x Speedup

kanna qed
1月3日
読了時間: 2分

"If accuracy is preserved, choosing 31x speed is the only logical choice."

Ghost Drift Research introduces a definitive departure from computational brute force.

Your Global Sum is a Waste of Time

—— O(N \log N) Audit-Ready FFT Replacement: A Proven 31x Performance Leap

The Bottleneck: The Hidden Tax of Global Sums

In large-scale simulations and data processing, "Global Sums" (all-to-all interactions) are a massive tax on performance. Calculating the "literal whole" at $O(N^2)$ is no longer a necessity—it’s an oversight.

The Solution: Ghost Drift’s "Finite Closure"

We have open-sourced a demo implementing the core pillar of Ghost Drift Theory: "Finite Closure."

By leveraging our proprietary Fejer-Yukawa (FY) window, you can transform $O(N^2)$ bottlenecks in physical simulations and complex calculations into high-efficiency FFT convolutions without losing mathematical integrity.

Empirical Evidence: Scaling from O(N^2) to O(N \log N)

This isn't just a tweak; it’s a fundamental shift in algorithmic complexity.

Speedup: 12.84s → 0.41s (31.3x Faster)
Precision Guarantee: Error strictly maintained within tolerance $\epsilon$ ($| \Delta \text{result} | \le 1e-6$)
Auditability: Complete verification via parameters and cryptographic hashes.

What is "Audit-Ready"?

A state where any third party can reproduce the exact same result given the same input, code, parameters, and fingerprint.

Audit Log Schema: [hash, params, epsilon, checksum, engine_version, seed, timestamp]

Key Concepts

Finite Closure: A methodology that takes an infinite "global" problem, fixes an error bound $\epsilon$ upfront, and "closes" the calculation into a finite window—fixing both the result and the accountability.
Fejer-Yukawa (FY) Window: The mathematical engine of finite closure. Unlike crude "cutoffs," it uses controlled decay to constrain errors strictly within $\epsilon$.

Technical Benchmarks

Complexity Shift: Global Sum (Direct): $O(N^2) \rightarrow$ FY window + FFT: $O(N \log N)$
Environment: $N=2,000,000$, float64, Apple M2 Max, NumPy/FFTW backend.

It’s Not an "Approximation." It’s a "Constraint."

Traditional optimizations often rely on "good enough" approximations. Ghost Drift is different. We fix the error bound $\epsilon$ first, then constrain the calculation within that finite window.

This is Audit-Ready Acceleration. In an era of black-box AI and opaque quantum computing, we deliver accountability in the language of mathematics. If the hash matches, the drift is accounted for.